We briefly review the classical approach to equilibrium and out of equilibrium statistical mechanics of long range interacting systems, for which the energy is not additive, and emphasize some new results. At equilibrium, we explain the thermodynamic consequences of the lack of additivity, like the generic occurrence of statistical ensemble inequivalence and negative specific heat. We then present a recent new classification of phase transitions and ensemble inequivalence in systems with long range interactions, and note a number of generic situations that have not yet been observed in any physical systems. Out of equilibrium, we show that algebraic temporal correlations or anomalous diffusion may occur in these systems, and can be explained using usual statistical mechanics and kinetic theory.
